Newform orbit 600.6.a.c

• Label: 600.6.a.c
• Level: $600$
• Weight: $6$
• Character orbit: 600.a
• Self dual: yes
• Analytic conductor: $96.230$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$600 = 2^{3} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	600.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$96.2302918878$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q - 9 * q^3 + 160 * q^7 + 81 * q^9 - 596 * q^11 + 122 * q^13 + 1078 * q^17 + 796 * q^19 - 1440 * q^21 + 1088 * q^23 - 729 * q^27 + 46 * q^29 - 4952 * q^31 + 5364 * q^33 + 6114 * q^37 - 1098 * q^39 - 6 * q^41 + 24116 * q^43 - 13480 * q^47 + 8793 * q^49 - 9702 * q^51 - 20598 * q^53 - 7164 * q^57 - 46756 * q^59 - 9602 * q^61 + 12960 * q^63 + 17404 * q^67 - 9792 * q^69 + 26568 * q^71 - 75450 * q^73 - 95360 * q^77 + 50472 * q^79 + 6561 * q^81 - 33236 * q^83 - 414 * q^87 + 133194 * q^89 + 19520 * q^91 + 44568 * q^93 + 42878 * q^97 - 48276 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

Label  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

0

0

−9.00000

0

0

0

160.000

0

81.0000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$5$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	600.6.a.c		1
5.b	even	2	1		120.6.a.f	✓	1
5.c	odd	4	2		600.6.f.a		2
15.d	odd	2	1		360.6.a.a		1
20.d	odd	2	1		240.6.a.g		1
40.e	odd	2	1		960.6.a.t		1
40.f	even	2	1		960.6.a.a		1
60.h	even	2	1		720.6.a.i		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.6.a.f	✓	1	5.b	even	2	1
240.6.a.g		1	20.d	odd	2	1
360.6.a.a		1	15.d	odd	2	1
600.6.a.c		1	1.a	even	1	1	trivial
600.6.f.a		2	5.c	odd	4	2
720.6.a.i		1	60.h	even	2	1
960.6.a.a		1	40.f	even	2	1
960.6.a.t		1	40.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7} - 160$ acting on $S_{6}^{\mathrm{new}}(\Gamma_0(600))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T + 9$
$5$	$T$
$7$	$T - 160$
$11$	$T + 596$